I Introduction
Let be the
dimensional vector space over the finite field
of elements with characteristic . For any vector (also, called word) , the Hamming weight of is defined to be the number of nonzero coordinates, i.e.,For integers , a linear code is a dimensional linear subspace of . The minimum distance of is the minimum Hamming weight among all nonzero vectors in , i.e.,
A linear code is called an linear code if has minimum distance . A wellknown tradeoff between the parameters of a linear code is the Singleton bound which states that
An code is called a maximum distance separable (MDS) code if . An important class of MDS codes are affine ReedSolomon codes and projective ReedSolomon codes, which will be our main object of study in this paper.
Let be an linear code over . The error distance of any word to is defined to be
where
is the Hamming distance between words and . The error distance plays an important role in the decoding of the code. The maximum error distance
is called the covering radius of . The words achieving maximum error distance are called deep holes of the code. Given a word , the word where and has the same error distance as . Therefore, it is natural as well as convenient, to study equivalence classes of deep holes of under the equivalence relation for and .
We recall the definition of affine and projective ReedSolomon codes.
Definition I.1.
Fix a subset , which is called the evaluation set. Let . For any integer , the generalized (affine) RS code of length , dimension and scale vector over is defined to be
It is easy to check that the minimum distance of this code is , and thus is an MDS code. For the case , we write for short. For any , the RS code is (monomially) equivalent to . Thus, we only consider throughout this paper. When , we write for short.
For any word
, by Lagrange interpolation, there is a unique polynomial
of degree at most such thatClearly, if and only if . We also say that is defined by the polynomial . One can easily show (see [9, Theorem 5.1]): for any , we have the inequality
It follows that if , then . One obtains the covering radius of all affine RS codes. Namely, we have the following
Proposition I.2.
The covering radius of all affine RS codes with parameters is .
Deciding deep holes of a given code is much harder than the covering radius problem, even for affine RS codes. The deep hole problem for affine RS codes was studied in [2, 3, 8, 9, 10, 11, 12, 13, 14]. As noted above, words with are deep holes of affine . Based on numerical computations, Cheng and Murray [3] conjectured that the converse is also true if .
Conjecture I.3 ([3]).
For (and if is even), a word is a deep hole of if and only if .
This conjecture remains open, but has been proved if either by [16] or by [7]. In particular, the conjecture is true for prime fields. The aim of this paper is to investigate deep holes of projective ReedSolomon codes. This turns out to be significantly more difficult.
Definition I.4.
The projective ReedSolomon (PRS) code is defined to be
where and is the coefficient of the term of degree of .
In the literature, the PRS code is also called doubly extended RS code. It is easy to see that the dual code of is the PRS code . As for RS codes above, we introduce the following notation: We represent a vector in by where is defined by a polynomial of degree at most :
and is arbitrary. It is easy to see that represents a codeword of if and only if
Affine RS codes of length are obtained by puncturing PRS codes on the last coordinate, and further puncturing yields all affine RS codes. Although they look very similar, the PRS code is much more difficult. For instance, the covering radius of the PRS code is already unknown. A consequence [5] of the MDS conjecture implies the following covering radius conjecture for projective ReedSolomon codes.
Conjecture I.5.
For , the covering radius of is except when is even, in which case the covering radius is .
Remark I.6.
The Conjecture I.5 is equivalent to a wellknown conjecture in finite geometry that the points of a ‘normal rational curve’ (in short ) in (the points of represented by the columns of the matrix of (1)) form a ‘complete arc’ except when is even and . For a proof of this equivalence see [7, §4] or [5, Theorem 2.4]. Similarly, Conjecture I.3 is equivalent to the conjecture (see [7, §3]) that for , any arc in with points on a NRC must have all its points on the NRC, except when is even and .
It is shown in [15][7] that the deep hole Conjecture I.3 also implies the above covering radius conjecture. Since the deep hole conjecture is known to be true if by [16] or if by [7], we deduce that the covering radius conjecture is true if or if . There are many other cases that the covering radius conjecture is known to be true, see [7], including the two simplest nontrivial cases and which will be used later in this paper. The next major problem is the deep hole problem for .
Problem I.7.
Let . Assume that the covering radius conjecture is true. Construct all deep holes of .
This problem is very difficult and wide open. As shown in [7], classifying deep holes of for even is equivalent to the difficult problem of classifying hyperovals of . In [15], a class of deep holes of is obtained, which are defined by polynomials of degree , if .
Theorem I.8 (Deep holes of degree [15]).
Let be a prime power and let be an integer such that if or if . Suppose . The words
are distinct deep hole classes of .
The fact that these words have error distance also occurs in older literature. In [6], this fact was used to show that the covering radius of is at most .
Remark I.9.
The authors of [6] claim in their Theorem 2, that the covering radius of not only but also all MDS codes of length (except when is even and ) is . While this result is unlikely to be false, they prove this theorem only for PRS codes, and hence the proof does not hold for general MDS codes of length . Moreover the proof for is also incorrect as it implicitly assumes the validity of the Conjecture on completeness of normal rational curves, mentioned in Remark I.6.
The subgroup of consisting of Hamming isometries of is the group of monomial matrices (i.e. a monomial matrix is a product of a permutation matrix and a diagonal matrix). For a linear code , the subgroup of monomial matrices which carry to itself is the automorphism group of the code , denoted Aut. Given a deep hole of , and , clearly and have the same error distance. Thus acts on the set of deep holes of . Let denote the group of scalar matrices. It follows that the group acts on the set of equivalence classes of deep holes of . As shown in [1, Theorem 2.2], for a PRS code , the group Aut is isomorphic to the projective general linear group . Applying these automorphisms to the deep hole classes of Theorem I.8, we get the following new classes of deep holes.
Theorem I.10.
Let be a prime power and let be an integer such that if or if . Suppose . The set of words
represents distinct classes of deep holes of except when , odd in which case this set represents only distinct classes of deep holes.
Below, we give another simple but very effective method, embedding the code to MDS supercodes, to control the error distance so that a new class of deep holes for can be constructed explicitly. We define to be the set of nonzero vectors in whose first nonzero entry is .
Theorem I.11 (Construction from irreducible quadratic polynomials).
Let be a prime power and let be an integer such that if or if . Let . Suppose . Let be a monic irreducible quadratic polynomial. For any which are not zero simultaneously, the rational function generates a deep hole of . More precisely, the set of words
represents distinct classes of deep holes of except when , odd in which case this set represents only distinct classes of deep holes.
Since the covering radius conjecture is true for , we get:

the covering radius of the code for odd is .

for any prime power and , the covering radius of the code is .
For the first case, in the work [7], one of the present authors showed that there are in total deep holes classes of , and these are represented by
The method in [7] was to characterize those points of which do not represent syndromes of deep holes of . In this work, we obtain a different proof of this result by showing that that the deep hole classes in Theorem I.11 also represent all the deep hole classes of .
Theorem I.12.
Let be odd. The deep hole classes in Theorem I.11 are all the deep holes of .
The main result of this paper is the following complete classification of deep holes for .
Theorem I.13.
When , there are other deep hole classes for , and the classification problem becomes increasingly more difficult as decreases. The code , which seems to be the next most accessible case, will be treated in a forthcoming work.
Ii Covering Radii and New Deep Holes for PRS codes
In this section, we study the covering radii and deep holes for PRS codes. Recall that the code has a generator matrix:
(1) 
It is easy to check that any minor of is nonzero, and hence is an MDS code. For the case , the PRS code is nothing but the repetition code generated by . In this case, one can easily show that the covering radius is and the deep holes are permutations of the element multiset , where is arbitrary.
For the case , the proof of Theorem II.1 in [15] shows that the covering radius of is and the deep hole classes are .
For the case , one can show that the covering radius of is and the deep hole classes are represented by the vectors of weight with entries in .
With the boundary cases removed, we can then assume that .
Iia Covering radii of PRS codes
Although the covering radius of RS codes is always , it seems a little surprising that the covering radius of PRS codes is unknown in general. The above boundary cases suggest that the covering radius of can have two possibilities , which is indeed the case. Another example in [4] is the PRS code over with generator matrix
The code has minimum distance and covering radius . This example suggests may have covering radius , two smaller than the minimum distance . This leads to Conjecture I.5. In [5], Dür proved that the covering radius of is provided that there is no MDS code extending the code by one digit. Thus Conjecture I.5 is equivalent to the conjecture that there is no MDS code extending the code by one digit, except when is even and . A partial list of known for which this conjecture is known to be true is given in [7, Proposition 4].
In [15] we studied the relationship between the covering radius Conjecture I.5 for PRS codes and the deep hole Conjecture I.3.
Theorem II.1 ([15]).
Let be an odd prime power. Assume that Conjecture I.3 is true. Then the covering radius of is .
Note that the above theorem holds for even : if Conjecture I.3 is true, then the covering radius of is for all . The proof is the same as the proof in [15].
Corollary II.2 ([15]).
Let be a power of the prime . Assume either or (and if ). Then the covering radius of is .
IiB Automorphisms of PRS codes
As observed in the §I, the automorphism group of a linear code acts on the set of its deep holes. Thus, we first begin with some notation and facts about automorphisms of PRS codes. For any integer and , we define:
(2) 
Using this notation, the generator matrix of can be written as
where and is an ordering of . Let denote the projective space , i.e. the set of equivalence classes of under the equivalence relation for and . For we denote its class in . The map identifies with . Let denote the group where is the group of scalar matrices. The action of on gives rise to an action of on . In particular acts on in the following way: for with image , and , we have . As usual, this means . Let be the permutation matrix defined by
Clearly is a group homomorphism, and thus
(3) 
There is a group homomorphism from to (see [1, §2] for details) with the property that the induced homomorphism from to is injective and satisfies for all . If then the th entry of is the coefficient of in the polynomial . For example th column of is
(4) 
It follows that for each such there is a diagonal matrix such that the monomial matrix satisfies the property
The matrix is defined by
if and
if It is clear from this definition that
(5) 
Since
and has full rank , it follows that is a group homomorphism. Thus . This together with (3) gives:
(6) 
We recall that the matrix is a parity check matrix for . Given a word , its syndrome is Syn. We observe that if is a noncodeword for a linear code and is a word equivalent to (where and ), then Syn. Conversely, if Syn, then . The relevance of syndromes of deep holes of is brought out by the next definition and lemma.
Definition II.3.
Let denote the set of projective syndromes of deep holes of , i.e. the image under of the syndromes of deep holes of .
Lemma II.4.
The map from the set of deep hole classes of to that takes the deep hole class represented by to is a bijective map.
The action of on words is and the action of on Syn is . We note that there are induced actions of the group on equivalence classes of noncodewords, and on the set of projective syndromes of noncodewords of . The next lemma shows that the bijective map of Lemma II.4 is equivariant under the action of on the source and target spaces of the map, i.e. .
Lemma II.5.
For and we have
In particular if represents a deep holes class of , and then .
We end this section with a lemma that we will need in the next section. Denote
(7) 
We will also use the same symbol for the vector .
Lemma II.6.
For , the stabilizer of under the action of is as follows.

The group if and is odd.

The group if and .

The group if and .

The whole group if and is even.
And is in orbit of if and only if . In case , the stabilizer of is the group .
Proof:
By (4), for any , we have
In order to determine when this equals , we consider the cases and separately.
If , then the first and last components imply , i.e. either or . In the former case which equals if and only if . If , then
. This proves the assertions 1) and 2).
In case , then in case , and when , using the fact that , we can write . This equals if and only if . This proves the assertions 3) and 4).
If , then for . If , then it is clear from (4) that, for every in the orbit of the last entry of . In particular, is not in the orbit of . Also, for and , we have
Thus stabilizes if and only if and . In other words the stabilizer of is the group . ∎
IiC Deep holes of PRS codes
A further question is to find all the deep holes of the PRS codes. On the opposite side, the following result shows that a large class of vectors are not deep holes.
Theorem II.7 ([15]).
If , denote . There are positive constants and such that if
then for any , is not a deep hole of .
On the positive side, we have the constructions of new deep holes of PRS codes given by Theorems I.10 and I.11. Theorem I.8 gives deep hole classes
(8) 
Using the wellknown result
it easily follows that the projective syndrome of is , which again shows (using Lemma II.4) that the deep hole clases in (8) are distinct. As shown in the previous section, all words in the orbit of these deep hole classes under are also deep holes. The deep hole classes thus obtained (excluding those in of Theorem (8) itself) are the classes given in Theorem I.10.
Theorem i.10.
Let be a prime power and let be an integer such that if or if . Suppose
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